PRINCIPLES OF NUMBER. 181 



Similarly, just as in logic 



triangle or square square or triangle, 

 or generally A I B = B I A, 



so in quantity 2 + 3 = 3 + 2, 



or generally x + y = y + x. 



The symbol I is not identical with + , but it is so far 

 analogous. 



How far, now, is it true that mathematical symbols 

 obey the law of simplicity expressed in the form 



or the example 



Round round = round \ 



Apparently there are but two numbers which obey this 

 law ; for it is certain that 



x x x x 

 is true only in the two cases when x= i or o. 



In reality all numbers obey the law, for 2x2 = 2 is not 

 really analogous to AA = A. According to the definition 

 of a unit already given, each unit is discriminated from 

 each other in the same problem, so that in 2' x 2 ', the 

 first two involves a different discrimination from the 

 second two. I get four kinds of things, for instance, if I 

 first discriminate ' heavy and light ' and then ' cubical and 

 spherical,' for we now have the following classes 

 heavy, cubical. light, cubical. 

 heavy, spherical. light, spherical. 

 But suppose that my two classes are in both cases 

 discriminated by the same difference of light and heavy, 

 then we have 



hea,vy heavy = heavy, 

 heavy light = o, 

 light heavy = o, 

 light light = light. 



In short, twice two is two unless we take care that the 

 second two has a different meaning from the first But 



